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To find the inverse of the function \( f(x) = \sqrt[3]{1 - x} \), we need to solve for \( x \) in terms of \( y \) and then interchange \( x \) and \( y \).

### Step-by-Step Solution

1. **Start with the function:**
   \[
   y = \sqrt[3]{1 - x}
   \]

2. **Isolate the cube root:**
   \[
   y^3 = 1 - x
   \]

3. **Solve for \( x \):**
   \[
   x = 1 - y^3
   \]

4. **Interchange \( x \) and \( y \) to find the inverse function:**
   \[
   y = 1 - x^3
   \]

5. **Write the inverse function:**
   \[
   f^{-1}(x) = 1 - x^3
   \]

### Final Answer

The inverse function is:
\[
f^{-1}(x) = 1 - x^3
\]

Would you like more details on this solution or have any other questions?

### Related Questions

1. How do you verify if two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. How do you find the domain and range of an inverse function?
4. What is the graphical relationship between a function and its inverse?
5. How does the process of finding an inverse change for different types of functions (e.g., quadratic, exponential)?

### Tip

To find an inverse function, remember to switch \( x \) and \( y \) after solving for \( y \) in terms of \( x \). This helps to represent the inverse function correctly.

Given the one-to-one function f(x) = ³√(1 - x), find f⁻¹(x).

Learn how to find the inverse of the function f(x) = ³√(1 - x) with a detailed step-by-step solution. This problem involves inverse functions, algebraic manipulation, and the properties of one-to-one functions. Suitable for students in Grades 10-12.

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